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2.3: The ASA and AAS Theorems


In this section we will consider two more cases where it is possible to conclude that triangles are congruent with only partial information about their sides and angles,

Suppose we are told that ( riangle ABC) has (angle A = 30^{circ}, angle B = 40^{circ}), and (AB =) 2 inches. Let us attempt to sketch ( riangle ABC). We first draw a line segment of 2 inches and label it (AB), With a protractor we draw an angle of (30^{circ}) at (A) and an angle of (40^{circ}) at (B) (Figure (PageIndex{1})). We extend the lines forming (angle A) and (angle B) until they meet at (C). We could now measure (AC, BC), and (angle C) to find the remaining parts of the triangle.

Let ( riangle DEF) be another triangle, with (angle D = 30^{circ}), (angle E = 40^{circ}), and (DE =) 2 inches. We could sketch ( riangle DEF) just as we did ( riangle ABC), and then measure (DF, EF), and (angle F) (Figure (PageIndex{2})). It is clear that we must have (AC = DF), (BC = EF), and (angle C = angle F), because both triangles were drawn in exactly the same way, Therefore ( riangle ABC cong riangle DEF).

In ( riangle ABC) we say that (AB) is the side included between (angle A) and (angle B). In ( riangle DEF) we would say that DE is the side included between (angle D) and (angle E).

Our discussion suggests the following theorem:

Theorem (PageIndex{1}): ASA or Angle-Side-Angle Theorem

Two triangles are congruent if two angles and an included side of one are equal respectively to two angles and an included side of the other.

In Figure (PageIndex{1}) and (PageIndex{2}), ( riangle ABC cong riangle DEF) because (angle A, angle B), and (AB) are equal respectively to (angle D), (angle E), and (DE).

We sometimes abbreviate Theorem (PageIndex{1}) by simply writing (ASA = ASA).

Example (PageIndex{1})

In ( riangle PQR), name the side included between

  1. (angle P) and (angle Q).
  2. (angle P) and (angle R).
  3. (angle Q) and (angle R).

Solution

Note that the included side is named by the two letters representing each of the angles. Therefore, for (1), the side included between (angle P) and (angle Q) is named by the letters (P) and (Q) -- that is, side (PQ). Similarly for (2) and (3).

Answer: (1) (PQ), (2) (PR), (3) (QR).

Example (PageIndex{2})

For the two triangles in the diagram

  1. write the congruence statement,
  2. give a reason for (1),
  3. find (x) and (y).

Solution

(1) From the diagram (angle A) in ( riangle ABC) is equal to (angle C) in ( riangle ADC). Therefore, "(A)" corresponds to "(C)". Also (angle C) in ( riangle ABC) is equal to (angle A) in ( riangle ADC). So "(C)" corresponds to "(A)". We have

(2) (angle A, angle C), and included side (AC) of ( riangle ABC) are equal respectively to (angle C), (angle A), and included side (CA) of ( riangle CDA). ((AC = CA) because they are just different names for the identical line segment, We sometimes say (AC = CA) because of identity.) Therefore ( riangle ABC cong riangle CDA) because of the ASA Theorem ((ASA = ASA)).

Summary:

(egin{array} {ccrclcl} {} & & {underline{ riangle ABC}} & & {underline{ riangle CDA}} & & {} { ext{Angle}} & & {angle BAC} & = & {angle DCA} & & { ext{(marked = in diagram)}} { ext{Included Side}} & & {AC} & = & {CA} & & { ext{(identity)}} { ext{Angle}} & & {angle BCA} & = & {angle DAC} & & { ext{(marked = in diagram)}} end{array})

(3) (AB = CD) and (BC = DA) because they are corresponding sides of the congruent triangles. Therefore (x = AB = CD = 12) and (y = BC = DA = 11).

Answer:

(1) ( riangle ABC cong riangle CDA).

(2). (ASA = ASA): (angle A, AC, angle C) of ( riangle ABC = angle C), (CA), (angle A) of ( riangle CDA).

(3) (x = 12), (y = 11).

Let us now consider ( riangle ABC) and ( riangle DEF) in Figure (PageIndex{3}). (angle A) and (angle B)

of ( riangle ABC) are equal respectively to (angle D) and (angle E) of ( riangle DEF), yet we have no information about the sides included between these angles, (AB) and (DE), Instead we know that the unincluded side BC is equal to the corresponding unincluded side (EF). Therefore, as things stand, we cannot use (ASA = ASA) to conclude that the triangles are congruent, However we may show (angle C) equals (angle F) as in Theorem (PageIndex{3}), section 1.5 ((angle C = 180^{circ} - (60^{circ} + 50^{circ}) = 180^{circ} - 110^{circ} = 70^{circ}) and (angle F = 180^{circ} - (60^{circ} + 50^{circ}) = 180^{circ} - 110^{circ} = 70^{circ})). Then we can apply the ASA Theorem to angles Band (C) and their included side (BC) and the corresponding angles (E) and (F) with included side EF. These remarks lead us to the following theorem:

Theorem (PageIndex{2}) (AAS or Angle-Angle-Side Theorem)

Two triangles are congruent if two angles and an unincluded side of one triangle are equal respectively to two angles and the corresponding unincluded side of the other triangle ((AAS = AAS)).

In Figure (PageIndex{4}), if (angle A = angle D), (angle B = angle E) and (BC = EF) then ( riangle ABC cong riangle DEF).

Proof

(angle C = 180^{circ} - (angle A + angle B) = 180^{circ} - (angle D + angle E) = angle F). The triangles are then congruent by (ASA = ASA) applied to (angle B). (angle C) and (BC) of (angle ABC) and (angle E, angle F) and (EF) of ( riangle DEF).

Example (PageIndex{3})

For two triangles in the diagram

  1. write the congruence statement,
  2. give a reason for (1),
  3. find (x) and (y).

Solution

(1) ( riangle ACD cong riangle BCD).

(2) (AAS = AAS) since (angle A, angle C) and unincluded side (CD) of (angle ACD) are equal respectively to (angle B, angle C) and unincluded side (CD) of ( riangle BCD).

(egin{array} {ccrclcl} {} & & {underline{ riangle ACD}} & & {underline{ riangle BCD}} & & {} { ext{Angle}} & & {angle A} & = & {angle B} & & { ext{(marked = in diagram)}} { ext{Angle}} & & {angle ACD} & = & {angle BCD} & & { ext{(marked = in diagram)}} { ext{Unincluded Side}} & & {CD} & = & {CD} & & { ext{(identity)}} end{array})

(3) (AC = BC) and (AD = BD) since they are corresponding sides of the congruent triangles. Therefore (x = AC = BC = 10) and (y = AD = BD). Since (AB = AD + BD = y + y = 2y = 12), we must have (y = 6).

Answer

(1) ( riangle ACD cong riangle BCD)

(2) (AAS = AAS): (angle A, angle C, CD) of ( riangle ACD = angle B, angle C, CD) of ( riangle BCD).

(3) (x = 10), (y = 6).

Example (PageIndex{4})

For the two triangles in the diagram

  1. write the congruence statement,
  2. give a reason for (1),
  3. find (x) and (y).

Solution

Part (1) and part (2) are identical to Example (PageIndex{2}).

(3):

(egin{array} {rcl} {AB} & = & {CD} {3x - y} & = & {2x + 1} {3x - 2x - y} & = & {1} {x - y} & = & {1} end{array}) and (egin{array} {rcl} {BC} & = & {DA} {3x} & = & {2y + 4} {3x - 2y} & = & {4} end{array})

We solve these equations simultaneously for (x) and (y):

Check:

Answer:

(1) and (2) same as Example (PageIndex{2}).

(3) (x = 2), (y = 1).

Example (PageIndex{5})

From the top of a tower Ton the shore, a ship Sis sighted at sea, A point (P) along the coast is also sighted from (T) so that (angle PTB = angle STB). If the distance from (P) to the base of the tower (B) is 3 miles, how far is the ship from point Bon the shore?

Solution

( riangle PTB cong riangle STB) by (ASA = ASA). Therefore (x = SB = FB = 3).

Answer: 3 miles

Historical Note

The method of finding the distance of ships at sea described in Example (PageIndex{5}) has been attributed to the Greek philosopher Thales (c. 600 B.C.). We know from various authors that the ASA Theorem has been used to measure distances since ancient times, There is a story that one of Napoleon's officers used the ASA Theorem to measure the width of a river his army had to cross, (see Problem 25 below.)

Problems

1 - 4. For each of the following (1) draw the triangle with the two angles and the included side and (2) measure the remaining sides and angle,

1. ( riangle ABC) with (angle A = 40^{circ}), (angle B = 50^{circ}), and (AB = 3) inches,

2. ( riangle DEF) with (angle D = 40^{circ}), (angle E = 50^{circ}), and (DE = 3) inches,

3. ( riangle ABC) with (angle A = 50^{circ}), (angle B = 40^{circ}), and (AB = 3) inches,

4. ( riangle DEF) with (angle D = 50^{circ}), (angle E = 40^{circ}), and (DE = 3) inches.

5 - 8. Name the side included between the angles:

5. (angle A) and (angle B) in ( riangle ABC).

6. (angle X) and (angle Y) in ( riangle XYZ).

7. (angle D) and (angle F) in ( riangle DEF).

8. (angle S) and (angle T) in ( riangle RST).

9 - 22. For each of the following

(1) write a congruence statement for the two triangles,

(2) give a reason for (1) (SAS, ASA, or AAS Theorems),

(3) find (x), or (x) and (y).

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23 - 26. For each of the following, include the congruence statement and the reason as part of your answer:

23. In the diagram how far is the ship S from the point (P) on the coast?

24. Ship (S) is observed from points (A) and (B) along the coast. Triangle (ABC) is then constructed and measured as in the diagram, How far is the ship from point (A)?

25. Find the distance (AB) across a river if (AC = CD = 5) and (DE = 7) as in the diagram.

26. that is the distance across the pond?


Difference Between ASA and AAS

Geometry is fun. Geometry is all about shapes, sizes, and dimensions. Geometry is the kind of mathematics that deals with the study of shapes. It is easy to see why geometry has so many applications that relate to the real life. It is used in everything – in engineering, architecture, art, sports, and much more. Today, we will discuss triangle geometry, specifically triangle congruence. But first, we need to understand what it means to be congruent. Two figures are congruent if one can be moved onto the other in such a way that all their parts coincide. In other words, two figures are called congruent if they are the same shape and size. Two congruent figures are one and the same figure, in two different places.

It’s true than triangle congruence is the basic building block for many geometrical concepts and proofs. Triangle congruence is one of the most common geometrical concepts in High school studies. One major concept often overlooked in teaching and learning about triangle congruence is the concept of sufficiency, that is, to determine the conditions which satisfy that two triangles are congruent. There are five ways to determine if two triangles are congruent, but we are going to discuss only two, that is, ASA and AAS. ASA stands for “Angle, Side, Angle”, while AAS means “Angle, Angle, Side”. Let’s take a look at how to use the two to determine if two triangles are congruent.


Triangle Congruence Postulates

Proving two triangles are congruent means we must show three corresponding parts to be equal.

From our previous lesson, we learned how to prove triangle congruence using the postulates Side-Angle-Side (SAS) and Side-Side-Side (SSS). Now it’s time to look at triangles that have greater angle congruence.

Angle-Side-Angle

And as seen in the figure to the right, we prove that triangle ABC is congruent to triangle DEF by the Angle-Side-Angle Postulate.

Angle-Angle-Side

And as seen in the accompanying image, we show that triangle ABD is congruent to triangle CBD by the Angle-Angle-Side Postulate.

As you will quickly see, these postulates are easy enough to identify and use, and most importantly there is a pattern to all of our congruency postulates.

Can you can spot the similarity?

Yep, you guessed it. Every single congruency postulate has at least one side length known!

And this means that AAA is not a congruency postulate for triangles. Likewise, SSA, which spells a “bad word,” is also not an acceptable congruency postulate.

We will explore both of these ideas within the video below, but it’s helpful to point out the common theme.

You must have at least one corresponding side, and you can’t spell anything offensive!

Knowing these four postulates, as Wyzant nicely states, and being able to apply them in the correct situations will help us tremendously throughout our study of geometry, especially with writing proofs.

So together we will determine whether two triangles are congruent and begin to write two-column proofs using the ever famous CPCTC: Corresponding Parts of Congruent Triangles are Congruent.


Triangle Congruences. Not!

A typical proof using triangle congruence will use three steps to set up the three congruent triangle parts (several may be givens), a fourth step invoking a triangle congruence theorem, followed by a CPCF ( Congruent Parts of Congruent Figures are congruent ) invocation to relate additional congruent triangle parts. This kind of proof is very similar to those using transitivity in that regard and lend themselves nicely to the two column format. However, outside geometry most proofs are written in paragraph style. Our text advises against including givens to cut down on thoughtless ritual. (Thus they deduce conclusions instead of making statements.) However, as a visual learner, I tend to disagree with the authors on this one. It can often be a useful way to organize what you know, making it easier to fill in what you don't. Thus we will be flexible in format and advise students to try a variety of approaches until they find what suits them. In this chapter the two column proof reigns supreme, however.

Below we will discuss the three triangle non-congruences of AAA, and SSA=ASS further. First we will discuss the four triangle congruences of SSS, SAS, SAA (which is the same as and is usually referred to as AAS), and ASA.

SSS is more formally known as the Side-Side-Side Triangle Congruence Theorem (or maybe Edge-Edge-Edge Triangle Congruence Theorem).

If in two triangles the three sides are pairwise congruent, then the triangles are congruent.

As with much of our textbook, it proves this using transformations (reflections preserve distance and the Kite Symmetry Theorem). This is important because it differs from Euclid's faulty superposition development. It also differs from other modern rigorous developments which use SAS as a postulate (Hilbert, Birkhoff). Whichever route you take to develop YOUR geometry, you should also be able to convince yourself using only a compass and straight-edge that SSS always yields congruence. The two triangles might have opposite orientation, but they will still be congruent. As implied by the faulty development of Euclid on this score, the proof of these triangle congruence theorems is more involved than the proofs we expect you to be able to write. However, we do expect you to be able to follow the proofs given. The SsA Triangle Congruence Theorem is the longest in our text and does not appear in many texts, including Euclid's Elements .

We spoke earlier about the 3-4-5 triangle being a right triangle. Of course, not all 3-4-5 triangles are going to be congruent because someone might use 3 attometers, 3 miles, or even 3 light-years. However, because of the Pythagorean Theorem these are all right triangles. (Personally, I have reservations about both attometers and light-years due to quantization of space-time and general relativity.) This is a fundamental property that given the three sides of a triangle, you have fixed the angles. This relates as well to the fact that triangles are rigid . Rigidity is an important property in the functionality of objects like doors, rafters, and gates.

SAS is more formally known as the Side-Angle-Side Triangle Congruence Theorem. Be sure the angle you are using is BETWEEN the two sides you are using. If sides AB and BC are used, angle B is the included angle. Order is important and is implied by the order the letters are specified.

If in 2 triangles 2 sides and the included angle are pairwise congruent, then the triangles are congruent.

AAS is more formally known as the Angle-Angle-Side Triangle Congruence Theorem. The side used here is opposite the first angle.

If in 2 triangles 2 angles and a non-included side are pairwise congruent, then the triangles are congruent.

ASA is more formally known as the Angle-Side-Angle Triangle Congruence Theorem. The side used here is BETWEEN the two angles you are using. If angle A and angle B are used, side AB is the included side.

If in 2 triangles 2 angles and included sides are pairwise congruent, then the triangles are congruent.
ASA Triangle AAS Triangle

If in two triangles two angles are pairwise congruent, then the triangles are similar .

There is NO SSA Triangle Triangle Congruence Theorem. (Although SSA and ASS are equivalent, please avoid the latter spelling, even though it fairly represents the situation should you invoke it.) This is what we refer to as the ambiguous case or SSA condition. You can think of condition as being like a disease. Please refer to the diagram above and note the following. Angle A is fixed (given). Side length AB is fixed (given). Side length BC is also fixed (given). However, there are two possibilities for C as indicated by where a circle centered at B intersects line AC. One is denoted C a and the other C o . C a results in an acute angle at C whereas C o results in an obtuse angle at C. (The opposite type of angle is formed at B, thus the triangle is always nonacute.) As long as BC is longer than the minimum distance between B and AC and shorter than AB, two triangles are possible. However, if BC is longer than AB only one triangle is possible (see SsA below). If BC is exactly equal to the minimum distance between B and AC, then only one triangle, a right triangle, is possible (see HL below). If BC is less than the minimum distance between B and AC, then no triangle is possible.

Just like there can be zero, one, or two solutions to a quadratic equation, there can be zero, one, or two triangles corresponding to a given SSA triple. This deeper connection is made more explicit by examining the quadratic nature of the Law of Cosines , a generalization of the Pythagorean Theorem.

Law of Cosines: In triangle ABC with sides of length a , b , and c : c 2 = a 2 + b 2 - 2 ab cos C .

Side a of length a =BC is opposite A , side b of length b =AC is opposite B , and side c of length c =AB is opposite C . Notice how the Law of Cosines as stated is symmetric in a and b —they can be interchanged with the same result. Which angle/side is used is also arbitrary, so we could just as well have written it as: a 2 = b 2 + c 2 - 2 bc cos A or b 2 = a 2 + c 2 - 2 ac cos B . Since the cos 90°=0, in a right triangle the Pythagorean Theorem results. Traditionally, in a right triangle, angle C is right and side c is then the hypotenuse. Thus the boxed statement of the law of cosines has additional merit.

If in two triangles two sides and the angle opposite the longer of the two sides are pairwise congruent, then the triangles are congruent.

Second, if BC is exactly equal to the minimum distance between B and line AC, then angle C is a right angle. BC and AC are then legs of this right triangle and AB is the hypotenuse. Notice how if BC were slightly longer, two triangles would result and it if were slightly shorter no triangle is possible. This is commonly referred to as the HL Triangle Congruence Theorem.

If in two right triangles the hypotenuse and a leg are pairwise congruent, then the triangles are congruent.

Other texts note that the SSA condition guarantees congruence if the congruent angles are nonacute ( i.e. right or obtuse). With information at the end of this chapter (Law of Sines, etc. ) it will be seen that these are equivalent. Note also how the HL Congurence Theorem is also a subset of SAS for right angles.

With the AAS triangle congruence theorem we can now prove the converse of the Isosceles Triangle Base Angles Theorem or the converse of pons asinorum .

If two angles of a triangle are congruent, then the sides opposite these angles are congruent.

Consider the kite ABCD at right. Note how triangles BCF and DCE both have the region AECF in their interiors. They thus overlap . Angle C is in common to both. Thus if we are given two additional pieces of information we may be able to prove these triangles congruent. You might consider what two pieces are required. Diagonals in regular polygons present similar situations.

The question often comes up as to how many triangles are formed by the diagonals in a polygon, perhaps regular. The answers for the triangle (1) and quadrilateral (8) border on trivial, but get rather involved for the pentagon, hexagon, etc. If you can find the value for a pentagon, this link will give you the answer for n > 5. Specifically, the sequence known as A006600 is for regular polygons and the sequence A005732 is for cyclic (inscribable in a circle) n -gons. Curiously, for n odd they are the same, whereas for n even they differ. This paper explains why.

There are shapes other than triangles which we can use to cover a region. When using general shapes the term tesselate is used. A tesselation uses a fundamental region to completely cover (or tile ) a plane such that no holes are found. This fundamental region is repeated via the various isometries (translation, rotation, reflection, glide reflection), hence they are all congruent. A regular polygon will tesselate only if the angle measure evenly divides 360°. Thus only the following regular polygons tesselate: triangles (60°), squares (90°), and hexagons (120°). Any triangle and any quadrilateral will tesselate because you can arrange them so that angles summing to 360° surround each vertex. However, only a few pentagons will tesselate and several new types were only recently discovered. Hexagons are likewise restricted.

M. C. Escher (1898-1972) was a recent Dutch artist who often utilized tesselations and other mathematical concepts, such as perspective in his works. The following link has a nice set of links to various tesselation sites and this one has a java applet useful for online study. A couple of these have contest deadlines in the March 28-April 1 range. Tesselmania demo. Tessel mania examples. Tess 1.51.

  1. Both pair of opposite sides are pairwise congruent
  2. Both pair of opposite angles are pairwise congruent and
  3. Diagonals intersect at their midpoints ( i.e. bisect each other).
  1. One pair of sides is both parallel and congruent or
  2. Both pairs of opposite sides are congruent or
  3. The diagonals bisect each other or
  4. Both pairs of opposite angles are congruent.

An angle is an exterior angle iff it forms a linear pair with an interior angle of a [convex] polygon.

Note: there are alternate definitions of exterior angle which may make just as much sense, but which violate the UCSMP limitation of angles being less than or equal to 180º. Please be clear which definition we are using.

Since an exterior angle of a triangle forms a linear pair with an interior angle, and that interior angle is supplementary to the sum of the other two interior angles, the following two theorems should be fairly self-evident. Please note that we refer to an exterior angle not the exterior angle since there are two possible exterior angles at each vertex.

The measure of an exterior triangle angle is equal to the sum of the measures of the interior angles at the other two vertices ( Exterior Angle Theorem ).

The measure of an exterior triangle angle is greater than the measure of either of the interior angles at the other two vertices ( Exterior Angle Inequality ).

This chapter concludes with two theorems about the lengths of sides in triangles. Specifically, the longer sides are opposite the larger angles. The text uses two theorems ( Unequal sides/angles Theorems ), and many nots to make this statement. This is actually a specific case of the Law of Sines given below. For nonobtuse triangles, since sine is a monotonic (always increasing or always decreasing) function between 0° and 90° it is easy to see how this works. For obtuse triangles one must also rely on the unequal angle theorem above and have deeper insight into the sin function.

Law of Sines: In triangle ABC with sides of length a , b , and c :
sin A/ a = sin B/ b = sin C/ c     or     a /sin A = b /sin B = c /sin C.

These concepts are also often referred to as the Hinge Theorem . This basically states that given two fixed length sides in a triangle, the length of the third side will increase as the angle opposite increases, just like the set of triangles described by a hinge.


4.08 Congruent Triangles- ASA and AAS

So far, you have learned to prove triangles are congruent using the SSS and SAS postulates. You will learn two more methods now.

Included Sides

Before we learn these methods, you need to learn a new concept. The concept is called included side.

An included side is the side of a triangle that is between two angles. In other words, the included side has to be touching both angles.

The included side of angles B and C is side BC becaus BC touches both angles B and C.

Let's see if you can pick out the included side between the angles given.

Name the included side between the given angles.

ASA Theorem

Great job, now you can move on to the new theorem.

The ASA Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

The order of the letters ASA angle, side, angle helps you to remember that side has to be between the two angles.

Example: ASA Theorem

List the corresponding congruent parts of the triangles. Is this enough information to prove the triangles congruent? If so, write the congruence statement and the method used to prove they are congruent.

Solution: First we will list all given corresponding congruent parts.

Now we have enough information to state the triangles are congruent. Since MO and OM are the corresponding included sides, △PMO ≅ △NOM by ASA Theorem .

Your Turn

List the corresponding congruent parts of the triangles. Is this enough information to prove the triangles congruent? If so, write the congruence statement and the method used to prove they are congruent.

Answer: △BCE by ASA Theorem

AAS Theorem

Here is the last theorem you will need to learn about proving two triangles are congruent.

The AAS Theorem states that if two angles and a nonincluded side of one triangle is congruent to two angles and a nonincluded side of another triangle, then the triangles are congruent.

The order of the letters AAS angle, angle, side helps you to remember that side is not between the two angles.

Example: AAA Theorem

List the corresponding congruent parts of the triangles. Is this enough information to prove the triangles congruent? If so, write the congruence statement and the method used to prove they are congruent.

Solution: First we will list all given corresponding congruent parts.

We have enough information to state the triangles are congruent. Since AC and EC are the corresponding nonincluded sides, △ABC ≅ △ ____ by ____ Theorem.

Answer: △EDC by AAS Theorem

Video

If you would like to see a video of someone explaining these concepts along with solving problems go to Brightstorm: ASA and AAS.


Does SAA prove congruence?

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not by itself prove congruence.

  1. If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.
  2. If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

Furthermore, is SAA a proof?

Proof of Theorem 6.19: SAA Congruence Theorem: If two angles of a triangle and a side opposite one of the two angles are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

What is SAA congruence postulate?

SAA postulate is one of the conditions for any two triangles to be congruent. Step 1: If two angles and the non-included side of one triangle is congruent to two angles and the non-included side of another triangle then the two triangles are congruent by SAA postulate.


Review

For questions 1-3, determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

  1. Figure (PageIndex<8>)
  2. Figure (PageIndex<9>)
  3. Figure (PageIndex<10>)

For questions 4-8, use the picture and the given information below.

Figure (PageIndex<11>)

Given: (overlineperp overline), (overline) is the angle bisector of (angle CDA)

  1. From (overlineperp overline), which angles are congruent and why?
  2. Because (overline) is the angle bisector of (angle CDA), what two angles are congruent?
  3. From looking at the picture, what additional piece of information are you given? Is this enough to prove the two triangles are congruent?
  4. Write a 2-column proof to prove (Delta CDBcong Delta ADB), using #4-6.
  5. What would be your reason for (angle Ccong angle A)?

For questions 9-13, use the picture and the given information.

Given: (overlineparallel overline), (overlinecong overline)

  1. From (overlineparallel overline), which angles are congruent and why?
  2. From looking at the picture, what additional piece of information can you conclude?
  3. Write a 2-column proof to prove (Delta LMPcong Delta OMN).
  4. What would be your reason for (overlinecong overline)?
  5. Fill in the blanks for the proof below. Use the given from above. Prove: (M) is the midpoint of (overline).

Determine the additional piece of information needed to show the two triangles are congruent by the given postulate.

  1. AAS Figure (PageIndex<13>)
  2. ASA Figure (PageIndex<14>)
  3. ASA Figure (PageIndex<15>)
  4. AAS Figure (PageIndex<16>)

Triangle Congruence Asa Aas And Hl Worksheet Answers

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With this quiz and attached worksheet you can evaluate how well you understand triangle congruence postulates. About congruent triangles worksheet with answer congruent triangles worksheet with answer. An equilateral triangle has three congruent sides.

One angle is obtuse and the other two angles are. Geometry worksheet congruent triangles asa and aas answers from triangle congruence worksheet 1 answer key source. Asa step by step lesson find the two triangles that have angle side angle going for them.

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Before look at the worksheet if you would like to know the stuff related to triangle congruence and similarity. Matching worksheet this makes you remember the labels found on the triangles. Asa and aas theorems.

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The length of one side can be found by. Worksheet given in this section is much useful to the students who would like to practice problems on proving triangle congruence. 4 triangle congruence chapter are you ready.

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There are five ways to find if two triangles are congruent, , , and. (side, side, side). stands for side, side, side and means that we have two triangles with all three sides equal. for example., , , and congruence date period state if the two triangles are congruent.

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Write the missing congruence property. Asa vs stands for angle, side, angle, while means angle, angle, side geometry is fun. geometry is all about shapes, sizes, and dimensions. geometry is the kind of mathematics that deals with the study of shapes. it is easy to see why geometry has so many applications that relate to the real life.

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Students will need to be able to identify the triangle congruence theorems (,, , , ). how is the concept or procedure explained or demonstrated instead of filling out a worksheet, students will visually see a picture and verbally answer the question. .

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Therefore, () example. worked examples of triangle congruence if two triangles have edges with the exact same lengths, then these triangles are congruent. the following video covers the and rules for congruent triangles. uses the, , , and postulates to find congruent triangles.

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Angle-side-angle () rule. angle-side-angle is a rule used to prove whether a given set of triangles are congruent. the rule states that if two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

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Angle-angle-side () rule. Asa (right triangle) warning no ass or postulate no cursing in math class a c b d e f not congruent there is no such thing as an postulate warning no postulate a c b d e f there is no such thing as an postulate not congruent.

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(side-side-side) (side-angle-side) (angle-side-angle) (angle-angle-side) (right angle-hypotenuse-side) let us learn them all in detail. (side-side-side) if all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by rule.

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Giving your teachers will get you an a, but giving your teachers sass will get you a one-way ticket to the principals office. suppose we have two triangles, and such that two sides of are congruent to two sides of. lets also suppose that the angles between these sides are congruent.

For the mini-lesson, students will create a three-tab organizer.on the front of the organizer, students will write on the first tab, on the second tab, and on the third tab. we discuss what the abbreviations stand for and then students identify which postulate can be used to prove the triangles from the do now are congruent.

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The following postulate, as well as the and similarity theorems, will be used in proofs just as, , , , and were used to prove triangles congruent. example using the similarity postulate. explain why the triangles are similar and write a similarity statement.

The student will be able to prove triangles congruent by, , or. additional learning objective(s) the student will be able to explain the differences of each type of transformation. the student will be able to combine transformations to prove triangles are congruent.

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Worksheet for and postulates -triangle congruence. topic congruence. helps understand congruence in terms of rigid motions we provide step-by-step solutions for every question. Using the tick marks for each pair of triangles, name the method, , , that can be used to prove the triangles congruent.

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Similar triangles will have congruent angles but sides of different lengths. congruent triangles will have completely matching angles and sides. their interior angles and sides will be congruent. A) b) c) not congruent d) ) a) b) c) not congruent d) ) a) not congruent b) c) d) ) a) b) not congruent c) d) state what additional information is required in order to know that the triangles are congruent for the reason given.

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Use the triangle congruence criteria and to determine that two triangles are congruent. practice worksheet for lesson. congruence a variation on is, which is angle-angle-side. recall that for you need two angles and the side between them. but, if you know two pairs of angles are congruent, then the third pair will also be congruent by the angle theorem.

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G.a, g.b. prove that triangles are congruent using the and congruence postulates. prove that triangles are congruent using the congruence postulate and congruence theorem. To access the worksheet and quiz at any time, simply pause the video. below, you also have the option of viewing the same on.

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, ,, notes.notebook, two triangles are congruent if one of the following are met. postulate (side side side) if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent postulate (side angle side).

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Triangle congruence postulates, , , ,. view course find a tutor next lesson. congruent triangles are triangles with identical sides and angles. the three sides of one are exactly equal in measure to the three sides of another. the three angles of one are each the same angle as the other.

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To find out, identify whether each pair of triangles is congruent by, , or. circle the letter that represents this characteristic. place the circled letters in the blanks at the bottom of the page above the corresponding problem numbers. (k) (b) (a) (r) (m) (l) (m) (e) (c) (t) (n).

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Back next example. Side, side), - (side, angle, side), - (angle, side, corner), - (angle, corner, side) - and (hypotenuses, foot) below, we discussed and theorems of congruent triangles. this means that the two triangles have all three sides in exactly the same way the ad we have to figure out the missing angles.

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Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.


Geometry: The AAS Theorem

You've accepted several postulates in this section. That's enough faith for a while. It's time for your first theorem, which will come in handy when trying to establish the congruence of two triangles.

  • Theorem 12.2: The AAS Theorem. If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the triangles are congruent.

Figure 12.7 will help you visualize the situation. In the following formal proof, you will relate two angles and a nonincluded side of ?AB to two angles and a nonincluded side of ?RST.

Figure 12.7 Two angles and a nonincluded side of ?ABC are congruent to two angles and a nonincluded side of ?RST.

The HL Theorem for Right Triangles

Whenever you are given a right triangle, you have lots of tools to use to pick out important information. For example, not only do you know that one of the angles of the triangle is a right angle, but you know that the other two angles must be acute angles. You also have the Pythagorean Theorem that you can apply at will. Finally, you know that the two legs of the triangle are perpendicular to each other. You've made use of the perpendicularity of the legs in the last two proofs you wrote on your own. Now it's time to make use of the Pythagorean Theorem.

  • Theorem 12.3: The HL Theorem for Right Triangles. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.

There are several ways to prove this problem, but none of them involve using an SSA Theorem. Your plate is so full with initialized theorems that you're out of room. Not to mention the fact that a SSA relationship between two triangles is not enough to guarantee that they are congruent. If you use the Pythagorean Theorem, you can show that the other legs of the right triangles must also be congruent. Then it's just a matter of using the SSS Postulate.

Figure 12.8 illustrates this situation. You have two right triangles, ?ABC and ?RST.

Figure 12.8 The hypotenuse and a leg of ?ABC are congruent to the hypotenuse and a leg of ?RST.

Tangled Knot

SSS, SAS, ASA, and AAS are valid methods of proving triangles congruent, but SSA and AAA are not valid methods and cannot be used. In Figure 12.9, the two triangles are marked to show SSA, yet the two triangles are not congruent. Figure 12.10 shows two triangles marked AAA, but these two triangles are also not congruent.

Figure 12.9 These two triangles are not congruent, even though two corresponding sides and an angle are congruent. The two congruent sides do not include the congruent angle!

Figure 12.10 These two triangles are not congruent, even though all three corresponding angles are congruent.


Watch the video: IXL K3: ASA and AAS Theorems Geometry (January 2022).